Investigation in Some Wave Properties of a Billiards Cue
Violins are not made – barrels and benches are made.
But violins like bread, grapes, and children are born and brought up.
Viner Brothers. A Visit to Minotaur
It seems that nothing can be easier than to take a good cue, reproduce all its properties – material, size, cross-sections, weight, and balance – and make hundreds or thousands of equally good cues. However, if you ask any respected craftsman, he will show you dozens of patterns for excellent cues, but will also tell you about the whole number of ordinary cues he made using the same patterns. What is the difference between professional game cues and mediocre home cues, and can this difference be described not only with subjective impressions, but also with specific physical values? The answer to this question can only be obtained through searching and comparing the characteristics which describe the interaction between the cue and the ball.
By itself this interaction is nothing but a blow, which is a process described in a specific field of the mechanics of solids and in thousands of theoretical and experimental studies. However, in spite of enormous attention to the subject, a satisfactory agreement between theory and experimental data has been received only recently. The current theory of elastic impact is a generalization of the theory by H. Hertz and the wave theory of Saint-Venant. Without going into details, it is worth noting that the interaction of the cue and the cue ball is not limited to their deformation only at the point of contact, but is also related to the distribution of longitudinal waves of compression and relief in them. It is the transmission of these waves that defines the time of contact between the cue and the cue ball as well as the fraction of energy transferred from the cue to the cue ball. In addition, the longitudinal load of the cue contributes to its loss of stability and lateral deformation and, consequently, to the generation of transverse oscillations which also absorb energy transmitted to the cue ball. We should not forget that any lateral deformation of the cue during the stroke can lead to significant deviations of the cue ball from its intended trajectory. All these factors suggest that in describing the mechanics of the cue, we need to considered its wave properties in addition to its geometry and weight characteristics.
In order to obtain quantitative characteristics describing wave properties of a cue, we need a measuring tool able to convert mechanical deformations into an electrical signal. The most affordable and suitable material for the role of a deformation sensor is piezoceramics which can induce electric charge on its surface under strain. On the basis of piezoceramics three piezoelectric sensors were produced for measuring oscillations at the tip of the shaft, at the butt joint and near the bumper of the butt.
A specific design feature of the sensor installed in the joint of shaft and butt was that the piezoelectric element was placed off the axis of the sensor, making it sensitive not only to longitudinal waves of compression and tension, but also to transverse oscillations of the cue.
For synchronizing the signals from the sensors and for determining the time when the tip of the cue contacted the cue ball, a contact group was used. The signals from the sensors and from the contact group were recorded with a digital oscilloscope and had a characteristic appearance.
The duration of the negative rectangular pulse in the lower part of the oscillogram corresponded to the time of the closed state of the contact group, that is, to the interaction time of the cue stick and the cue ball. This time was about 1.5 ms and showed a weak dependence on the force of impact. In the case of a strong impact, the interaction time of the cue stick and the cue ball was 1.3 ms, the time of an average impact amounted to 1.5 ms, and the time of a weak impact was around 1.8 ms.
Such weak dependence between the contact time and the impact force (in other words, between the time of the contact and the speed of the cue at the moment of impact) does not contradict theoretical concepts about elastic collision. Thus, for the elastic collisions of identical spheres there is a theoretically calculated and experimentally verified dependence between the contact time and the relative velocity of collision T~V-1/5.At the same time, similar results for the longitudinal impact of homogeneous rods showed the absence of any dependence between the contact time and the relative velocity of collision. Apparently, the weak dependence observed during the collision of a cue stick and a cue ball is to a greater extent related to the deformation of the tip, rather than to the longitudinal deformation of the cue stick and the cue ball. The results of the experiment aimed at determining the interaction time for the cue stick from the increased thickness of the tip are in favor of this assumption. Thus, two identical tips set one on the other made it possible to increase the contact time from 1.5 ms to 2.8 ms. In addition, the contact time became heavily dependent on the force of impact (the change of the duration of contact could be as high as 70 percent). Measuring the contact time when striking the ball with the cue without a tip could provide another confirmation for the above assumption.
As was mentioned above, the interaction of the cue and the cue ball is not limited only to the deformation of the tip, but depends on the distribution of longitudinal compression waves in the cue. If we divide the contact time into four parts, where the first part Тab and last part Тde correspond to the time of the deformation and restoration of the tip, the duration of the remaining two parts according to the theory of Saint-Venant, will be determined by the time needed for elastic waves to run from the tip of the shaft to the bumper of the butt and back, that is, Tbd=2L/C, where L is – дlength of the cue stick, and C is the speed of sound in the cue stick. If in this equation we substitute the length of cue L = 1600 mm and the speed of sound in cue C = 4600 m / s (below this value will be determined experimentally), this time will be about 0.7 ms. It is not difficult to guess that for the cue twice shorter, the overall time of its contact with the cue ball will decrease by 0.35 ms. For verifying this assertion, we compared the time of contact during the impact by the intact cue and by the cue shortened by almost half. For this purpose, the butt of the cue was replaced by a short handle of brass equal in weight to the butt.
The figures below show the oscillograms of synchronized signals from the contact group and the piezoelectric sensor installed in the joint of the butt and the shaft.
The left figure shows the impact of the intact cue stick on the cue ball (interaction time of 3.4 * 500 µs = 1.7 ms); the right figure shows the impact of a half-size cue stick (interaction time of 2.7 * 500 µs = 1.35 ms); the difference in contact time was exactly 0.35 ms. The force of impact on the cue ball in both cases was equal and was controlled by the amplitude of the piezoelectric sensor signal installed in the joint. The values of these amplitudes CH1max = 11,4 В, which corresponds to the impact energy of 1600 mJ according to the calibration of the piezoelectric sensor. These values are shown in the upper right corner of the oscillograms.
The process of calibrating the piezoelectric sensor was, in fact, the measurement of the maximum amplitude of the signal when the cue stick was dropped (with its tip down) on a concrete floor from a certain height. As can be seen from the graph, the output voltage of the sensor has a fairly linear dependence on the impact energy (impact force), and the sensitivity of the sensor was extremely high, so we even had to make it two times less sensitive.
Measurement of the impact energy can also make it possible to estimate the speed of the cue ball after the impact, since the impact energy of 8000 mJ (about two calories) corresponds to velocity of the cue ball of around 7.5 m / sec.
For defining the propagation velocity of longitudinal waves in the cue, we recorded synchronized signals from two piezoelectric sensors installed in the joint and in the bumper. The blue curve shows the signal from the piezoelectric sensor installed in the joint; red curve shows the signal from the sensor in the bumper; the distance between the sensors was 780 mm. Everything beyond the green line (± 200 µs, two cells) occurs in the cue after the cue ball had separated from the tip of the cue stick. The piezoceramic elements of both sensors have the same sensitivity that was specifically tested (the difference did not exceed 0.5 percent). Despite the same sensitivity, the amplitude of the signals from both sensors differed by 30 times (the vertical axes to the right and to the left of the green line have the scale of 60V and 2V respectively). This difference in amplitudes is related not so much to the attenuation of the signal over the length of the cue, but with structural features of sensors’ fixation.
If we divide the distance between the sensors (780 mm) by the time interval between the points (a) and (b), which equals 170 µs, we can obtain the velocity of propagation of the elastic wave of around 4600 m / sec. This velocity is typical for hardwood. If we determine the velocity on the basis of the interval of time between points (c) and (d), we obtain the velocity of 7800 m / sec. It is impossible to have such an enormous speed in wood. This discrepancy can be explained if we assume that the signals from the sensors undergo a strong distortion during the registration process. There can be many reasons for this distortion, for example, spurious capacitances and inductances. However, the main reason apparently lies in the discordance between the sensors and the oscilloscope. Piezoelectric sensor can be represented as a very small capacitance (a condenser), where the charge (and hence the voltage) appear during the deformation. If the sensor is not connected to anything, the charge and the voltage will remain at its output until the time when the load is removed from the sensor. If the sensor is connected to an oscilloscope with input resistance of 1 MΩ, the charge will fairly quickly flow through this resistance from one end of the sensor to the other, and the voltage on the sensor output will be zero. The time of charge flow, and thus the shape of the sensor response to the load will directly depend on the input resistence of the oscilloscope. This was confirmed by the changes in the time interval between points (c) and (e) when the input of the oscilloscope was bypassed (and thus its input resistance was decreased).
Special equipment makes it possible to match piezoelectric sensors and recording devices. Charge amplifiers with the integration of the current (based on the operational amplifier with input on fieldistors) are capable of detecting signals even with frequencies below 1 Hz.
On the basis of an operational amplifier KR544UD2 we made a similar device that made possible to simultaneously record signals from three piezoelectric sensors. The principle of the design becomes clear from this film: http://www.youtube.com/watch?v=B2Uhg6UMKmM . First fifteen seconds of the film show a signal from the piezoelectric sensor sent directly to the input of the oscilloscope. In this case the piezoelectric sensor responds to the compression with two bursts, one responding to the increase of the load, and the other responding to the withdrawal of the load. The second fifteen seconds of the film show a signal from a piezoelectric sensor sent to the input of the oscilloscope through a charge amplifier. In this case, the oscilloscope screen shows the “true” type of the load.
The testing of the measuring system consisting of piezosensor and a charge amplifier was carried out while measuring the lateral deformation of a cue stick. As we mentioned above, the lateral deformation of the cue can be provoked by axial load. The behavior of the cue to some extent is similar to the behavior of a long rod, which means that both for a rod and a cue we should distinguish deformations caused by the static and the dynamic axial loads.
We should first consider the static problem. The situation in which the rod (beam, cue, etc.), loaded along the axis of symmetry, begins to bend in the transverse direction, is called the loss of stability. Under the static load, this problem was first solved by Euler. He calculated the critical force Fcr at which the rod loses its stability. This force is called the Euler force. But the thing is that under a static load only one form of deformation – a single half-sinewave (C-shaped) is possible. Other forms of bending are possible under static load only if there are additional intermediate supports.
The problem of dynamic loss of stability for the rod was solved and experimentally verified by Lavrentiev and Shabat (The Problems of Hydrodynamics and their Mathematical Models, p. 364). The main conclusion of this work is that the dynamic (impact) load, in contrast to the static load, the deformation of the rod can take various forms. When load F, n times exceeding the Euler force Fcr is instantly applied to the rod, the rod bends not according to the sinusoidal waves with the number of halfwaves equal to the square root of n / 2 or the nearest whole number. If the rod collapses under the load, the number of fractures is also equal to this number. Depending on the initial shape of the transverse deformation in a rod (beam, cue, etc.) the corresponding oscillation mode is excited.
Such mode of excitation of transverse oscillations after the central hit of the cue against the cue ball was captured by means of piezoelectric sensor installed in the joint of shaft and butt and connected to an oscilloscope through a charge amplifier. Sharp peak in the beginning of the ocsillogram is a longitudinal compression wave on a given scale (the scale of 10 µs per 1 cm; the compression wave in the previous oscilograms was drawn on a scale of 250–500 µs per 1 cm). Periodic signal after the peak is nothing elso but damped transverse oscillations of the cue; the period of these oscillations is 34–36 µs, and their frequency, respectively, is 29–27 Hz.
If we compare the form of damped transverse oscillations with a strictly periodic signal, we may see a small adjustment of the oscillations in the initial part of the excitation, which appear to be related to a slight difference of the initial shape of the transverse deformation of the cue from one of the ideal cases shown in figures a, b and c.
Similar oscillations and their initial adjustment are observed in the cue if the cue lightly strikes the table with its tip. The frequency of these damped oscillations is about 26–27 Hz, and their period, respectively, is 35 µs, which well agrees with previous measurements. If we continue the comparison of transverse oscillations of the cue with the same oscillations of a long rod of a uniform cross-section, we can assume that the frequency of 27 Hz is not a single natural frequency, and therefore both for the rod and for the cue there should be infinitely many frequencies of transverse oscillations F1, F2,…, Fn,……
Measurement of the first few natural frequencies can be performed under artificial excitation of transverse oscillations of the cue under the influence of external periodic force (the so-called forced oscillations). For generating forced oscillations, the cue was fixed in the cantilever manner in a lathe chuck in a horizontal plane. Strong permanent magnet (neodymium magnet, NdFeB) was glued to the tip of the cue; an electrical coil was set under the magnet, which was connected through an amplifier to a generator of harmonic signals.
Varying magnetic field generated by the coil excited the transverse oscillations of a given frequency in the cue. When the oscillations were excited at frequencies equal to the natural frequencies of the cue, the resonance occured accompanied by a sharp increase in the amplitude of oscillations, which was measured with the piezoelectric sensor installed in a joint.
Four characteristic peaks of the oscillation amplitude correspond to first four natural frequencies of the cue. The frequency F1 was approximately 8 Hz, F2=26 Hz, F3=44 Hz и F4=94 Hz. . In turn, each of these frequencies corresponds to its transverse mode of oscillation. Large oscillation amplitude at frequencies F1 and F2 makes possible to visually observe the deformation of the cue stick at these frequencies. Thus, at the frequency of F1 only one nodal point in the place where the cue was fixed, is visible, which corresponds to the C-shaped oscillation mode. At the oscillation frequency F2 two nodal points are visible, which corresponds to the S-shaped oscillation mode. The first nodal point is located in the place where the cue was fixed, the second nodal point is located at a distance of 390 mm from the tip, dividing the cue at a ratio of 0.73 L / 0.27 L. In comparison with the oscillation of a homogeneous rod of uniforn cross-section, in the cue the position of the second nodal point is slightly shifted towards the jammed end.
The frequency of 45 Hz, corresponding to the third oscillation mode, has the smallest amplitude even when compared with the amplitude of the fourth oscillation mode. The thing is that at the third oscillation mode, one of the nodal points (the point where the oscillation amplitude is 0) practically coincides with the joint, that is, with the place where the piezoelectric sensor is located.
It is not hard to guess that the position of the joint at certain oscillation modes (for two-piece cue it is the second and the fourth modes) can in the same way coincide with the areas with a maximum amplitude of oscillations (antinodal points). Such close proximity of the joint even of small weight and these areas can lead to significant parasitic swinging of the cue and to the changes in the characteristics of impact. If the joint is set near the nodal point, its weight will only affect the balance of the cue.
In conclusion, I would like to say that this study does not pretend to be a complete description of the entire complex mechanics of the cue stick, but is only an attempt to measure some of its characteristics. A logical continuation of this work could be to measure the longitudinal and transverse deformations of the cue with three synchronized piezoelectric sensors, and to compare the results of these measurements for the cues of different designs.
 For more detail on these theories and experiments which prove them, see, for example, the study: Kilchevsky N.A. Teoriya soudareniya tverdykh tel. [The theory of the impact of solid bodies]. Kiev: Naukova dumka, 1969; and the survey by J.F. Bell, Eksperimental’nye osnovy mekhaniki deformiruemykh tverdykh tel. [Experimental foundations of the deformable solid mechanics]. 2 vols. Moscow: Nauka, 1984.
 Since the interaction force of the tip of the cue and the cue ball is not a constant value throughout the whole interaction time, it is more convenient to operate not with the value of force, but with such a value as impact energy.